R. Guénette

A new mixed finite element method for computing viscoelastic flows

Abstract A new mixed finite element method for computing viscoelastic flows is presented. The mixed formulation is based on the introduction of the rate of deformation tensor as an additional unknown. Contrary to the popular EVSS method [D. Rajagopalan, R.A. Brown and R.C. Armstrong, J. Non-Newtonian Fluid Mech., 36 (1990) 159], no change of variable is performed into the constitutive equation. Hence, the described method can be used to compute solutions of rheological models where the EVSS method does not apply. The numerical strategy uses a decoupled iterative scheme as a preconditioner for the GMRES algorithm. The stability and the robustness of the method are investigated on two benchmark problems: the 4:1 contraction flow problem and the stick-slip flow problem. Numerical results for the PTT [N. Phan-Thien and R.I. Tanner, J. Non-Newtonian Fluid Mech., 2 (1977) 353] and the Grmela [J. Grmela, J. Rheology, 33 (1989) 207] models show that our method is remarkably stable and cheap in computer time and memory.

NUMERICAL SIMULATION OF VISCOELASTIC FLOWS THROUGH A PLANAR CONTRACTION

Abstract In this study, three nonlinear rheological models consisting of the Giesekus, the FENE-P, and the Phan-Thien-Tanner model are used to simulate the flow of a viscoelastic fluid through a planar 4:1 contraction. Both stress and velocity fields are examined at different sections of the flow and the predictions of the numerical simulations are compared with the experimental results of L.M. Quinzani, R.C. Armstrong adn R.A. Brown, J. Non-Newtonian Fluid Mech., 52 (1994) 1–36. Overall, the numerical simulations allow a description of the essential features of the flow, and reproduce much of the experimental results with good accuracy. Excellent qualitative agreement between the numerical results and the experimental observations is reported. However, the agreement remains semi-quantitative especially for the first normal stress difference around the entry section of the flow. This is to be expected in view that the simulations were limited to only one-mode models.

Numerical analysis of the modified EVSS method

Abstract In this paper, we present a proof of the stability of a new mixed finite element method introduced recently by the authors [9] within the context of viscoelastic fluids. The mixed formulation is related to the EVSS (Elastic-Viscous-Split-Stress) method proposed by Rajagopalan et al. [14] and is based on the introduction of the rate of deformation tensor as an additional unknown. The proof applies to the Stokes flow and some linearized constitutive equations of slow viscoelastic flows. Existence and uniqueness of the continuous and the discrete problems are derived from a generalized Brezzi-Babuska theory. It is shown that no additional compatibility condition is required between the various variables except the usual one for the velocity and the pressure fields. This result allows to choose low order finite element for the stress. Several numerical experiments on the 4:1 contraction Stokes flow will be presented which will confirm the improved stability obtained with this new formulation.

On the discrete EVSS method

Abstract This paper is a completion of a previous paper written by two of the authors [M. Fortin, R Guenette, R. Pierre, Numerical analysis of the modified EVSS method, Comput. Methods. Appl. Mech. Engrg. 143 (1997) 79–95] concerning the DEVSS method, which is a discrete variant of the popular elastic-viscous-split-stress (EVSS) method introduced by [Rajagopalan, R.A. Brown, R.C. Armstrong, Finite element methods for calculation of steady viscoelastic flow using constitutive equations with a Newtonian viscosity, J. Non-Newtonian Fluid Mech. 36 (1990) 159–199] within the context of viscoelastic fluids. The method uses the rate of deformation tensor as an additional variable. In this paper, it is shown that a simple modification of the theoretical framework introduced in the first paper [M. Fortin, R Guenette, R. Pierre, Numerical analysis of the modified EVSS method, Comput. Methods. Appl. Mech. Engrg. 143 (1997) 79–95] allows the treatment of multi-mode constitutive equations of viscoelastic flows as well as that of the velocity gradient introduced as the additional variable instead of the rate of deformation tensor. Moreover, this general framework based on a generalized Brezzi–Babuska theory, allows different discretizations for the stress and the additional variable. The paper presents a general criteria insuring the stability of the method. Three particular mixed finite elements satisfying this criteria are studied. An error analysis is performed on Stokes and viscoelastic flows.

Investigation of the abrupt contraction flow of fiber suspensions in polymeric fluids

Abstract Numerical simulations of the flow of rigid fibres through a 4:1 planar contraction, and the predicted flow pattern and fiber orientation are presented. Entirely new is the examination of the nature of the suspending matrix which may consist of either a Newtonian fluid or a polymer melt. In the case of a polymer matrix three rheological models, the Phan-Thien–Tanner, FENE-CR, and Carreau models have been used to investigate the effects of shear-thinning and elasticity on the flow and the orientation of the fibers. The effects of inertia are neglected, and the governing equations for the flow field, polymer stress, and fiber orientation are coupled and simultaneously solved. A parametric study is used to explore the effects of different dimensionless parameters on the velocity field, the fiber orientation, the pressure drop, as well as the vortex size measured by the dimensionless reattachment length. We particularly focus on the role of the fibers aspect ratio, volume fraction, and interaction coefficient which measures the intensity of fiber interaction in the suspension. Furthermore, we evaluate and compare the results of four different closure approximations: the quadratic, linear, hybrid A and T, and natural closures.

Entry flow calculations using multi-mode models

Abstract In this study, two rheological models, namely the Giesekus and the Phan-Thien-Tanner models, are used to simulate the flow of a viscoelastic polymer solution through a planar 4:1 contraction. Both stress and velocity fields are examined at different sections of the flow. The predictions of the numerical simulations using one-mode as well as four-mode models are compared with the experimental results from the literature.

Micromechanical Modeling of Tensile Behavior of Short Fiber Composites

A simple micromechanical constitutive model is developed for short fiber reinforced composites (SFRC) undergoing damage. The model is based on the Carman and Reifsnider approach for the prediction of mechanical properties of discontinuous fiber reinforced composites. The composite is modeled by a distributed representative element composed of concentric circular cylinders using a general 3D configuration. The micromechanical model is used to evaluate the elastic properties of SFRC, by varying the orientation distribution of the fiber, the length distribution of the fiber and the fiber–fiber interaction phenomena. The composite is assumed to behave as linearly elastic in absence of any debonding of fiber from the matrix and in the fully debonded stage. The stress–strain behavior of molded composite materials and the debonding are modeled using the Hsueh model to estimate the debonding stress for misaligned fibers. A good agreement between calculated and experimental data was achieved.

An efficient hierarchical preconditioner for quadratic discretizations of finite element problems

Higher order finite element discretizations, although providing higher accuracy, are considered to be computationally expensive and of limited use for large-scale problems. In this paper, we have developed an efficient iterative solver for solving large-scale quadratic finite element problems. The proposed approach shares some common features with geometric multigrid methods but does not need structured grids to create the coarse problem. This leads to a robust method applicable to finite element problems discretized by unstructured meshes such as those from adaptive remeshing strategies. The method is based on specific properties of hierarchical quadratic bases. It can be combined with an algebraic multigrid (AMG) preconditioner or with other algebraic multilevel block factorizations. The algorithm can be accelerated by flexible Krylov subspace methods. We present some numerical results on the convection–diffusion and linear elasticity problems to illustrate the efficiency and the robustness of the presented algorithm. In these experiments, the performance of the proposed method is compared with that of an AMG preconditioner and other iterative solvers. Our approach requires less computing time and less memory storage. Copyright

Numerical Simulation of the Deformation of Drops under Simple Shear Flow

In this paper, we perform a numerical simulation of the deformation of a drop under simple shear flow. The drop and the fluid matrix can be Newtonian or viscoelastic fluids. For viscoelastic fluids, the Oldroyd-B model is used in a log-conformation formulation as in Fattal et al. /I/ and Guenette et al./2/. An anisotropic adaptive remeshing method is used and combined with a level set method for the computation of the free surfaces. Comparisons are made with similar simulations presented by Chung et al. /3/ and Chinyoka et al. /4/.